The interview was recorded on Thursday and Trump says that the letter was sent "yesterday", i.e. on Wednesday.
I have written a letter to them stating: "I hope you will negotiate, because if we have to go in militarily, it will be terrible", says Trump, according to AP.
The White House confirms that the goal is to get negotiations with Iran about the country's nuclear energy program.
"Would rather negotiate"
The message comes at the same time as both the US and Israel have warned that they will never allow Iran to develop nuclear weapons, which has led to fears of a military conflict with Tehran.
I would rather negotiate a deal. I'm not sure everyone agrees with me, but we could achieve a deal that would be just as good as if we won militarily, says Trump in the Fox interview.
But it's time now. Something will happen one way or another.
Iranian state media has referred to the Trump interview, but it is unclear whether the letter has reached the ayatollah.
A spokesperson for Iran's UN ambassador tells AFP that "no such letter has been received yet".
Iran's Foreign Minister Abbas Araghchi says to AFP that no new negotiations on nuclear energy are currently possible as long as Trump does not tone down his rhetoric.
We will not enter into direct negotiations with the US as long as they continue their approach with maximum pressure and threats, he says.
Highly enriched uranium
Iran has significantly increased its stockpile of highly enriched uranium recently. A report from the UN's atomic energy agency IAEA showed in February that Iran has almost 275 kilos of uranium enriched up to 60 percent, which is a short step from the 90 percent level needed to produce nuclear weapons.
Donald Trump unilaterally withdrew the US from an agreement with Tehran in 2015 during his first term as president. The agreement involved sanctions relief in exchange for limitations on Iran's nuclear program.
Since then, several warnings have been issued that Iran is on its way to producing nuclear weapons – something that the regime in Tehran denies.
